Fun fact- partially through their configuration and partially due to how humans and many other creatures bodies behave- if you get a quill stuck in you and do not remove it- as the wound tries to heal the barb on the quill often causes it to be pushed DEEPER into the flesh. This means untreated the wound will likely get worse and likely become infected and if left so may lead to death (primarily from sepsis and related infection.)
The barbs make quills hard to remove and very painful to do so. In general of your animal friend gets quilled you should NOT try to remove the quills yourself. You shouldn’t “cut” the quills either (a common urban legend) as that just increases the likelihood they’ll fracture or break and complicate things. As painful as the experience can be- oddly enough most dogs that are quilled do not seem to learn from the experience. Some may develop a fear or aversion and if lucky that will keep them away- but more often than not a dog that was quilled once if given the chance will get quilled again.
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